nation that results in a stationary time series. Expanding the linear combi-
nation and rearranging some terms, we have
(5.3)If the combination in Equation 5.3 must be stationary, the nonstationary
component must be zero, implying that , or the trend component
of one series must be a scalar multiple of the trend component in the other
series. Therefore, for two series to be cointegrated, the trends must be iden-
tical up to a scalar. In later chapters, we will rely on the Stock-Watson
model to establish links between arbitrage pricing theory and cointegration.
nnyztt=γyzn ntty z y z−=− +−γγεγε()()ttttOverview 79
HIGHLIGHTING THE POINT
This is an anecdote about an ingenious little kid. He was asked on a
test to say a few sentences about a cow. The poor lad knew only how
to say a few sentences about a tree. Thinking for a moment, the kid in
his first sentence tied the cow to a tree and then went on to talk about
the tree. The example is probably a little tongue in cheek. It is, how-
ever, true that there is a strong urge to relate the unknown to some-
thing familiar and enhance understanding through association.
To further underscore the point, it is worth mentioning that the
spirit of that approach actually forms the basis for formal proof tech-
niques in mathematics. Proof by induction, a technique attributed to
Cantor, relies on forming a series of logical relationships from the
most general to the most trivial. Proofs of NP completeness, used to
classify algorithms in the area of computational complexity theory (at-
tributed to Richard Karp), also rely on transformations to a known
problem. It is probably safe to say that in almost all fields of human
endeavor there is a common tendency to relate the intractable to some-
thing manageable and leverage existing knowledge to arrive at mean-
ingful conclusions.
We are no different from everyone else in this respect. When faced
with the prospect of having to work with nonstationary time series, we
immediately look for ways to construct portfolios that can be related
to stationary time series. The transformation to stationarity is typically
achieved using cointegration ideas and strict parity relationships.
Needless to say, this approach appears as a recurrent theme in the de-
sign of trading strategies across all asset classes.